Seconding @mican's recommendation for Affinity Propergation.

From the papar: ["L Frey, Brendan J., and Delbert Dueck. "Clustering by passing messages between data points." *science* 315.5814 (2007): 972-976."][1].

Its super easy to use via many packages.
Only one parameter -- *damping* and it works on anything you can define the pairwise similarity one. Which you can, by multiplying the Levenshtein distance by -1.

I threw together a quick example using the first paragraph of your question as input. In python3:

    import numpy as np #from numpy package
    import sklearn.cluster  # from sklearn package
    import distance #from distance package
    
    words = "YOUR WORDS HERE" #Replace this line
    lev_similarity = -1*np.array([[distance.levenshtein(w1,w2) for w1 in words] for w2 in words])
    
    affprop = sklearn.cluster.AffinityPropagation(affinity="precomputed", damping=0.5)
    affprop.fit(lev_similarity)
    for cluster_id in np.unique(affprop.labels_):
        exemplar = words[affprop.cluster_centers_indices_[cluster_id]]
        cluster = np.unique(words[np.nonzero(affprop.labels_==cluster_id)])
        cluster_str = ", ".join(cluster)
        print(" - *%s:* %s" % (exemplar, cluster_str))
        
        

Output was (exemplars in italics to the left of the cluster they are exemplar of):

 - *have:* chances, edit, hand, have, high
 - *following:* following
 - *problem:* problem
 - *I:* I, a, at, etc, in, list, of
 - *possibly:* possibly
 - *cluster:* cluster
 - *word:* For, and, for, long, need, should, very, word, words
 - *similar:* similar
 - *Levenshtein:* Levenshtein
 - *distance:* distance
 - *the:* that, the, this, to, with
 - *same:* example, list, names, same, such, surnames
 - *algorithm:* algorithm, alogrithm
 - *appear:* appear, appears


Running it on a list of [50 random first names][2]:

 - *Diane:* Deana, Diane, Dionne, Gerald, Irina, Lisette, Minna, Nicki, Ricki
 - *Jani:* Clair, Jani, Jason, Jc, Kimi, Lang, Marcus, Maxima, Randi, Raul
 - *Verline:* Destiny, Kellye, Marylin, Mercedes, Sterling, Verline
 - *Glenn:* Elenor, Glenn, Gwenda
 - *Armandina:* Armandina, Augustina
 - *Shiela:* Ahmed, Estella, Milissa, Shiela, Thresa, Wynell
 - *Laureen:* Autumn, Haydee, Laureen, Lauren
 - *Alberto:* Albertha, Alberto, Robert
 - *Lore:* Ammie, Doreen, Eura, Josef, Lore, Lori, Porter


Looks pretty great to me. (that was fun).

  [1]: http://www.sciencemag.org/content/315/5814/972.short
  [2]: http://listofrandomnames.com/