This is a Phase 1 Study of Eribulin Mesylate in Pediatric Participants With Recurrent or Refractory Solid Tumors (Excluding [Central Nervous System] CNS), Including Lymphomas

Inclusion Criteria * Participants must be \>=12 months and \<18 years of age at the time of study enrollment (Part A1). * Participants must be \>6 months and \<12 months of age at the time of study enrollment (Part A2). Participants will enroll one dose level behind the dose level at which participants in Part A1 are enrolling. * Participants with refractory or recurrent solid tumors or lymphomas, excluding CNS tumors, are eligible. Participants must have had histologic verification of malignancy at original diagnosis or relapse. Participants with primary CNS tumors, known CNS metastases, or a prior history of CNS metastases are not eligible. * Participants must have either measurable or evaluable disease. * Participants current disease state must be one for which there is no known curative therapy or therapy proven to prolong survival with an acceptable quality of life. * Karnofsky \>= 50% for participants \>16 years of age and Lansky \>=50 for participants less than or equal to (\<=)16 years of age. Participants who are unable to walk because of paralysis, but who are up in a wheelchair, will be considered ambulatory for the purpose of assessing the performance score. * Participants must have fully recovered from the acute toxic effects of all prior anticancer chemotherapy. 1. Myelosuppressive chemotherapy: At least 21 days after the last dose of myelosuppressive chemotherapy (42 days if prior nitrosourea). 2. Hematopoietic growth factors: At least 14 days after the last dose of a long-acting growth factor (example Neulasta) or 7 days for short-acting growth factor. For agents that have known adverse events occurring beyond 7 days after administration, this period must be extended beyond the time during which adverse events are known to occur. The duration of this interval must be discussed with the study chair. 3. Biologic (anti-neoplastic agent): At least 14 days after the last dose of a biologic agent. For agents that have known adverse events occurring beyond 14 days after administration, this period must be extended beyond the time during which adverse events are known to occur. The duration of this interval must be discussed with the study chair. 4. Immunotherapy: At least 42 days after the completion of any type of immunotherapy, example tumor vaccines. 5. Monoclonal antibodies: At least 3 half-lives of the antibody after the last dose of a monoclonal antibody. 6. X-ray telescope (XRT): At least 14 days after local palliative XRT (small port); At least 150 days must have elapsed if prior total body irradiation(TBI), craniospinal and/or entire spinal XRT or if \>=50% radiation of pelvis; At least 42 days must have elapsed if other substantial bone marrow (BM) radiation. 7. Stem Cell Infusion without TBI: No evidence of active graft versus host disease and at least 84 days must have elapsed after transplant or stem cell infusion. * Adequate Bone Marrow Function Defined as: 1. Peripheral absolute neutrophil count (ANC) \>=1000 per cubic millimeter (/mm\^3). 2. Platelet count \>=100,000/mm\^3 (transfusion independent, defined as not receiving platelet transfusions for at least 7 days prior to enrollment). 3. Hemoglobin (Hb) at least 8 gram per deciliter (g/dL) at baseline (blood transfusions are allowed during the screening period to correct Hb values less than 8 g/dL). All participants enrolled on the study must be evaluable for hematologic toxicity. * Adequate Renal Function Defined as: 1. Creatinine clearance or radioisotope glomerular filtration rate (GFR) \>=70 milliliter per minute (ml/min) per (/) 1.73 square meter (m\^2) or 2. A serum creatinine milligram per deciliter (mg/dL) based on age/gender as follows: 1. 6 months to \<1 year: male, 0.5; female, 0.5 2. 1 to \< 2 years: male, 0.6; female, 0.6 3. 2 to \< 6 years: male, 0.8; female, 0.8 4. 6 to \< 10 years: male, 1; female, 1 5. 10 to \< 13 years: male, 1.2; female, 1.2 6. 13 to \< 16 years: male, 1.5; female, 1.4 7. \>=16 years: male, 1.7; female, 1.4 The threshold creatinine values were derived from the Schwartz formula for estimating GFR (Schwartz et al., 1985) utilizing child length and stature data published by the Centers for Disease Control and Prevention (CDC). * Adequate Liver Function Defined as: 1. Bilirubin (sum of conjugated + unconjugated) \<=1.5 \* upper limit of normal (ULN) for age 2. serum glutamic-pyruvic transaminase (SGPT) (alanine transaminase \[ALT\]) \<=110 units per liter (U/L). For the purpose of this study, the ULN for SGPT is 45 U/L. 3. Serum glutamic-oxaloacetic transaminase (SGOT) (aspartate aminotransferase \[AST\]) \<= 125 U/L. For the purpose of this study, the ULN for SGOT is 50 U/L. 4. Serum albumin \>= 2 g/dL * Adequate Cardiac Function Defined as: 1. Shortening fraction of \>= 27% by echocardiogram, or ejection fraction of \>= 50% by gated radionuclide study 2. Corrected QT interval (QTc) \<= 480 millisecond (msec) Note: Participants with Grade 1 prolonged QTc (450-480 msec) at the time of study enrollment should have correctable causes of prolonged QTc addressed if possible (that is, electrolytes, medications). * All participants and/or their participants or legally authorized representatives must sign a written informed consent. Assent, when appropriate, will be obtained according to institutional guidelines. Participants must be willing to comply with all aspects of the protocol. * Participants with known human immunodeficiency virus (HIV) who have CD4+ T cell counts greater than or equal to 500 cells/m\^3 and who do not require antiretroviral therapy are eligible. Exclusion Criteria * Pregnant or breast-feeding women will not be entered on this study due to risks of fetal and teratogenic adverse events as seen in animal studies. Pregnancy tests must be obtained in girls who are post-menarchal. Males or females of reproductive potential may not participate unless they have agreed to use an effective double barrier contraceptive method for the entire period in which they are receiving protocol therapy and up to 6 months after treatment. * Concomitant Medications * Participants receiving corticosteroids who have not been on a stable or decreasing dose of corticosteroid for at least 7 days prior to enrollment are not eligible. * Participants who are currently receiving another investigational drug are not eligible. * Participants who are currently receiving other anticancer agents are not eligible. * Participants who are receiving cyclosporine, tacrolimus or other agents to prevent graft-versus-host disease post bone marrow transplant are not eligible for this trial. * Participants who are receiving drugs that prolong the QTc are not eligible. * Participants who have received prior therapy with eribulin mesylate are not eligible. * Participants with hypersensitivity to excipients of the study drug are not eligible. The excipients are ethanol, hydrochloric acid, sodium hydroxide and water for injection. * Participants who have a prior history of viral hepatitis (B or C) as demonstrated by positive serology (presence of antigens) or have an uncontrolled infection requiring treatment are not eligible. * Participants with greater than Grade 1 peripheral sensory neuropathy or greater than Grade 1 peripheral motor neuropathy graded according to the Modified ("Balis") Pediatric Scale of Peripheral Neuropathies are not eligible. * Cardiac Pathology * Participants with known congestive heart failure, symptomatic or left ventricle (LV) ejection fraction 50% or shortening fraction less than 27% are not eligible. * Participants with congenital long QT syndrome, bradyarrhythmias, or QTc greater than 480 msec are not eligible. * CNS Disease * Participants with primary CNS tumors are not eligible. * Participants with prior history of or known metastatic CNS disease involvement are not eligible. (Note: CNS imaging for participants without a known history of CNS disease is only required if clinically indicated). * Participants who have had or are planning to have the following invasive procedures are not eligible: * Major surgical procedure, laparoscopic procedure, open biopsy or significant traumatic injury within 28 days prior to enrollment. * Central line placement or subcutaneous port placement is not considered major surgery but must be placed at least 3 days prior to enrollment for external lines (with example, Hickman or Broviac) and at least 7 days prior to enrollment for subcutaneous port. * Core biopsy within 7 days prior to enrollment. * Fine needle aspirate within 7 days prior to enrollment. NOTE: For purposes of this study, bone marrow aspirate and biopsy are not considered surgical procedures and therefore are permitted within 14 days prior to start of protocol therapy. * Participants with known bone marrow involvement are not eligible. * Participants who have received a prior solid organ transplantation are not eligible. * Participants who in the opinion of the investigator may not be able to comply with the safety monitoring requirements of the study are not eligible.