From the General Introduction to the Liturgy of the Hours:
“245. For a public cause or out of devotion, except on solemnities, the Sundays of the seasons of Advent, Lent, and Easter, Ash Wednesday, Holy Week, the octave of Easter, and 2 November, a votive office may be celebrated, in whole or in part: for example, on the occasion of a pilgrimage, a local feast, or the external solemnity of a saint.”
(Documents on the Liturgy 1963-1979, Liturgical Press, Minnesota, 1982, ISBN 0-8146-1281-4, page 1127).
CathMass wrote “Now, there is an approved Votive Mass that can be used in honor of St. Anne which I know is used at Ste Anne de Beaupre in Canada and at St. Ann’s Basilica in Scranton PA.”
I doubt that this is a Votive Mass. The 2002 Roman Missal lists 19 Votive Masses in the contents section, (Missale Romanum, Libreria Editrice Vaticana, 2002, ISBN: 8820972719, page 1317). The saints with a Votive Mass are: Blessed Virgin Mary, John the Baptist, Joseph, Peter and Paul.
On 26 July the church has the memorial of “Joachim and Ann, parents of Mary”. The table of "General Norms for the Liturgical Year and the Calendar of 21 March 1969 has a “Table of Liturgical Days According to their Order of Precedence”. It includes: “4. … c. Solemnity of the title of a particular church.” (Documents on the Liturgy 1963-1979, Liturgical Press, Minnesota, 1982, ISBN 0-8146-1281-4, page 1165). So perhaps this is the basis for what happens at “St. Ann’s Basilica in Scranton PA.” etc.
The 2002 GIRM has “355. … c. On the weekdays in Ordinary Time, it is possible to choose either a weekday Mass, or the Mass of an optional memorial which happens to occur on that day, **or the Mass of any Saint listed in the Martyrology for that day ** , or a Mass for Various Needs, or a Votive Mass.”
If the Roman Martyrology lists St. Heliconis for 28 May, then the Opening Prayer at Mass today could be from the Common for Holy Women:
“Father, rewarder of the humble, you blessed St. Heliconis with charity and patience. May her prayers help us, and her example inspire us to carry our cross and to love you always. We ask this through our Lord Jesus Christ, your Son, who lives and reigns with you and the Holy Spirit, one God for ever and ever.”
(Roman Missal, Catholic Book Publishing Co., New York, 1985, page 821.)
Similarly for the Divine Office, following General Instruction to the Liturgy of the Hours, n. 244:
“On weekdays when an optional memorial is permitted, for a good reason the office of a saint listed on that day in the Roman Martyology, or in an approved appendix to it, may be celebrated in the same way as other memorials …”.