What does IHS on a Christian cross mean?

It is a monogram for the name of Jesus - see related link below:

Many confused and erroneous explanations of the meaning of the IHS monogram exist, most ascribing meanings to each letter as an acronym. However, IHS is not an acronym at all.

The monogram originated from abbreviations of the name 'Jesus' in ancient Greek script written by Hebrew scribes. The English 'Jesus' derives from the Hebrew name 'Yeshua' (יֵשׁוּעַ), a common alternative form of the name 'Joshua' (יְהוֹשֻׁעַ - Yehoshuah) from Hebrew "to rescue, deliver, save". Hebrew scribes translating the Old Testament into Greek (before the birth of Christ) transliterated 'Yeshua' to Greek Ἰησοῦς ("Iēsous") using the same root sound 'Yesh-' combined with the Greek nominative case suffix "-ous" later reduced to "Iesus". Later Roman scribes, rather than repeatedly writing this full name in their texts on the subject, abbreviated or contracted the root of the name to two capital Greek letters, 'IH' (iota - eta), and added their own 'S' for the nominative case suffix "-us" (masculine singular, second declension). This was very common for such repeated words as sacred names (nomina sacra) and there are many more such instances in Greek and Roman manuscripts from the period, including other early Christian monograms.

Thus יֵשׁוּעַ became Ἰησοῦς became IHS for JE-su-S

The IHS monogram gained popularity in the 12th century when St. Bernard insisted much on devotion to the Holy Name of Jesus, and 14th century Blessed John Colombini, founder of the Jesuati, often wore it on his breast. It became a symbol not unlike the Chi-Rho first popularized in the 4th century, and it was iconographic for St. Vincent Ferrer (d. 1419). St. Bernardine of Sienna (d. 1444) truly spread its use by brandishing it as a shield at sermons and having it inscribed on plaques and walls as a symbol of peace and piety. St. Ignatius of Loyola adopted the monogram as the emblem of the Jesuit Order (1541), incorporating a cross above the "H" and three nails below, surrounded by the sun.